(x-7)(x%2B35)(x%2B90)-67x%5E2.jpeg "(x-18)(x-7)(x+35)(x+90)-67x^2")
Factoring and Simplifying the Expression (x-18)(x-7)(x+35)(x+90)-67x^2
This expression involves a product of four binomials and a subtraction of a term with x². Let's break down how to approach simplifying and potentially factoring it:
1. Expanding the Binomials
The first step is to multiply the binomials together. This can be done in a systematic way, for example, using the distributive property:
- Expand (x-18)(x-7): (x-18)(x-7) = x² - 7x - 18x + 126 = x² - 25x + 126
- Expand (x+35)(x+90): (x+35)(x+90) = x² + 90x + 35x + 3150 = x² + 125x + 3150
Now, our expression becomes: (x² - 25x + 126)(x² + 125x + 3150) - 67x²
2. Expanding the Product
We need to multiply the two trinomials resulting from the previous step. This can be a bit tedious, but we can use the distributive property again, multiplying each term of the first trinomial by each term of the second trinomial.
This process will lead to a polynomial with terms up to the fourth power (x⁴).
3. Combining Terms
After expanding the product, we'll have a long polynomial with various terms. Combine the like terms (terms with the same power of x) to simplify the expression.
4. Potential Factoring
At this point, you might be able to factor the simplified polynomial further.
- Look for common factors among the terms.
- Consider using grouping techniques if applicable.
- Try to factor the polynomial into binomials or trinomials if possible.
5. The Difficulty of Factoring
It's important to note that this type of expression might not always factor easily or completely. There are cases where the resulting polynomial is irreducible, meaning it cannot be factored further.
In summary, the process involves expanding the binomials, multiplying the resulting trinomials, simplifying by combining like terms, and finally attempting to factor the resulting polynomial. While the steps are straightforward, the process can be quite complex and may not always lead to a fully factored expression.